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Type Theory and Programming
, 1994
"... This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an im ..."
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This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents MartinLof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.
Autarkic Computations in Formal Proofs
 J. Autom. Reasoning
, 1997
"... Formal proofs in mathematics and computer science are being studied because these objects can be verified by a very simple computer program. An important open problem is whether these formal proofs can be generated with an effort not much greater than writing a mathematical paper in, say, L A ..."
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Cited by 21 (1 self)
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Formal proofs in mathematics and computer science are being studied because these objects can be verified by a very simple computer program. An important open problem is whether these formal proofs can be generated with an effort not much greater than writing a mathematical paper in, say, L A T E X. Modern systems for proofdevelopment make the formalization of reasoning relatively easy. Formalizing computations such that the results can be used in formal proofs is not immediate. In this paper it is shown how to obtain formal proofs of statements like Prime(61) in the context of Peano arithmetic or (x + 1)(x + 1) = x 2 + 2x + 1 in the context of rings. It is hoped that the method will help bridge the gap between the efficient systems of computer algebra and the reliable systems of proofdevelopment. 1. The problem Usual mathematics is informal but precise. One speaks about informal rigor. Formal mathematics on the other hand consists of definitions, statements and proo...
Higherorder Annotated Terms for Proof Search
 THEOREM PROVING IN HIGHER ORDER LOGICS: 9TH INTERNATIONAL CONFERENCE, TPHOLS’96
, 1996
"... A notion of embedding appropriate to higherorder syntax is described. This provides a representation of annotated formulae in terms of the difference between pairs of formulae. We define substitution and unification for such annotated terms. Using this representation of annotated terms, the proof s ..."
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Cited by 20 (3 self)
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A notion of embedding appropriate to higherorder syntax is described. This provides a representation of annotated formulae in terms of the difference between pairs of formulae. We define substitution and unification for such annotated terms. Using this representation of annotated terms, the proof search guidance technique of rippling can be extended to higherorder theorems. We illustrate this by several examples based on an implementation of these ideas in Prolog.
Extensional Equality in Intensional Type Theory
 In LICS 99
, 1999
"... We present a new approach to introducing an extensional propositional equality in Intensional Type Theory. Our construction is based on the observation that there is a sound, intensional setoid model in Intensional Type theory with a proofirrelevant universe of propositions and rules for  and t ..."
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Cited by 20 (10 self)
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We present a new approach to introducing an extensional propositional equality in Intensional Type Theory. Our construction is based on the observation that there is a sound, intensional setoid model in Intensional Type theory with a proofirrelevant universe of propositions and rules for  and types. The Type Theory corresponding to this model is decidable, has no irreducible constants and permits large eliminations, which are essential for universes. Keywords. Type Theory, categorical models. 1. Introduction and Summary In Intensional Type Theory (see e.g. [11]) we differentiate between a decidable definitional equality (which we denote by =) and a propositional equality type (Id ( ; ) for any given type ) which requires proof. Typing only depends on definitional equality and hence is decidable. In Intensional Type Theory the type corresponding to the principle of extensionality Ext x2:(x) f;g2(x2:(x)) ( x2 Id (x) (f(x); g(x))) ! Id x2:(x) (f; g) is not...
Algorithms for Equality and Unification in the Presence of Notational Definitions
 Types for Proofs and Programs
, 1998
"... this paper we investigate the interaction of notational definitions with algorithms for testing equality and unification. We propose a syntactic criterion on definitions which avoids their expansion in many cases without losing soundness or completeness with respect to fi fficonversion. Our setting ..."
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Cited by 19 (11 self)
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this paper we investigate the interaction of notational definitions with algorithms for testing equality and unification. We propose a syntactic criterion on definitions which avoids their expansion in many cases without losing soundness or completeness with respect to fi fficonversion. Our setting is the dependently typed calculus [HHP93], but, with minor modifications, our results should apply to richer type theories and logics. The question when definitions need to be expanded is surprisingly subtle and of great practical importance. Most algorithms for equality and unification rely on decomposing a problem
AlgorithmSupported Mathematical Theory Exploration: A Personal View and Strategy
, 2004
"... We present a personal view and strategy for algorithmsupported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottomup mathematical invention, the algorit ..."
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Cited by 18 (5 self)
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We present a personal view and strategy for algorithmsupported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottomup mathematical invention, the algorithmic generation of conjectures from failing proofs for topdown mathematical invention, and the possibility to program new reasoners within the logic on which the reasoners work ("metaprogramming").
Faster Proof Checking in the Edinburgh Logical Framework
 In 18th International Conference on Automated Deduction
, 2002
"... This paper describes optimizations for checking proofs represented in the Edinburgh Logical Framework (LF). The optimizations allow large proofs to be checked eciently which cannot feasibly be checked using the standard algorithm for LF. The crucial optimization is a form of result caching. To f ..."
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Cited by 17 (3 self)
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This paper describes optimizations for checking proofs represented in the Edinburgh Logical Framework (LF). The optimizations allow large proofs to be checked eciently which cannot feasibly be checked using the standard algorithm for LF. The crucial optimization is a form of result caching. To formalize this optimization, a path calculus for LF is developed and shown equivalent to a standard calculus.
A TwoLevel Approach towards Lean ProofChecking
, 1996
"... We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle t ..."
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Cited by 16 (4 self)
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We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our twolevel approach is illustrated by some examples developed in Lego.
Un Calcul De Constructions Infinies Et Son Application A La Verification De Systemes Communicants
, 1996
"... m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to ..."
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Cited by 16 (0 self)
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m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w
Mathematical Vernacular and Conceptual Wellformedness in Mathematical Language
 Proceedings of the 2nd Inter. Conf. on Logical Aspects of Computational Linguistics, LNCS/LNAI 1582
, 1998
"... . This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive developmen ..."
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Cited by 14 (9 self)
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. This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem proving technology. The idea of semantic wellformedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of wellformedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV. 1 Introduction By mathematical vernacular (MV), we mean a mathematical and n...